15-May-2025 20:47:35

rref_test():
  MATLAB/Octave version 6.4.0
  rref() computes the reduced row echelon form (RREF) of
  a matrix.

  The RREF is susceptible to roundoff error.
  With that in mind, it can be used to determine:
  * the rank of a square matrix;
  * the solution of a nonsingular square linear system;
  * the inverse of a nonsingular square matrix.
  * the determinant of a square matrix.
  * the column rank of a rectangular matrix;
  * the independent columns of a rectangular matrix
  * if a linear system is consistent;

  Test 1, just get the RREF.

  Matrix A:
   1   3   0   2   6   3   1
  -2  -6   0  -2  -8   3   1
   3   9   0   0   6   6   2
  -1  -3   0   1   0   9   3

  A has rank 3

  RREF of A:
   1.0000   3.0000        0        0   2.0000        0        0
        0        0        0   1.0000   2.0000        0        0
        0        0        0        0        0   1.0000   0.3333
        0        0        0        0        0        0        0

  Test 2, look for inconsistency.

  Matrix A:
   1  -2   3  -1
   3  -6   9  -3
   0   0   0   0
   2  -2   0   1
   6  -8   6   0
   3   3   6   9
   1   1   2   3

  A has rank 3

  RHS b1:

     1:            1
     2:            2
     3:            3
     4:            4
     5:            5
     6:            5
     7:            7

  [A;b1] has rank 4
  Linear system A*x=b1 is INCONSISTENT.

  RHS b2:

     1:            4
     2:           12
     3:            0
     4:            2
     5:           12
     6:            9
     7:            3

  [A;b2] has rank 3
  Linear system A*x2=b2 is CONSISTENT.

  Solution of A*x2=b2:

     1:            1
     2:            0
     3:            1
     4:            0

  Compare b2 and A*x2:

       1:              4             4
       2:             12            12
       3:              0             0
       4:              2             2
       5:             12            12
       6:              9             9
       7:              3             3

  Test 3, inverse.

  Matrix A:
    5    7    6    5
    7   10    8    7
    6    8   10    9
    5    7    9   10

  Estimated inverse of A:
   68.0000  -41.0000  -17.0000   10.0000
  -41.0000   25.0000   10.0000   -6.0000
  -17.0000   10.0000    5.0000   -3.0000
   10.0000   -6.0000   -3.0000    2.0000

  Product A_inv * A:
   1.0000e+00   7.4607e-14  -3.1974e-14   6.3949e-14
   5.3291e-15   1.0000e+00   8.8818e-15   3.5527e-15
  -1.7764e-15  -2.8422e-14   1.0000e+00  -3.5527e-15
   4.6629e-15   1.7542e-14   5.5511e-15   1.0000e+00

  Test 4, determinant.

  Matrix A:
    5    7    6    5
    7   10    8    7
    6    8   10    9
    5    7    9   10

  Estimated determinant of A = 1:

  MATLAB det(A) = 1

  Test 5, independent columns.

  Matrix A:
    1    1    2    3
    2    3    4    9
    3    0    6    0
    4    2    8    0
    5    6   10    6
    6    3   12    6
    7    1   14    2

  Number of independent columns is 3

  Independent columns of A:
   1   1   3
   2   3   9
   3   0   0
   4   2   0
   5   6   6
   6   3   6
   7   1   2

rref_test():
  Normal end of execution.

15-May-2025 20:47:35